Mathematics - 2022-12-18

Ted Shifrin

00:02

You haven't proved anything yet.

What's the difference between using $f$ and $-f$, etc.?

Derivative

I don't know

Ted Shifrin

Well, you can't just write down the function and quit.

You have to prove some stuff about it, don't you?

Derivative

yes, but please do help me out. Is that the correct function?

Ted Shifrin

For starters, what is the zero set of your function?

Derivative

And what about the thing I said about not using orientability?

Ted Shifrin

00:04

How do you know whether you've used it or not, if you don't try to prove anything?

Derivative

ok, I'll just shut up and try to do the thing

Ted Shifrin

Just do your construction with two overlapping intervals on the $x$-axis in the plane.

Derivative

00:24

god this multivariate stuff is the most not-fun math I have ever studied

Dhruv

00:41

@TedShifrin Are you algebraic geometer?

Dhruv

00:51

@TedShifrin

Ted Shifrin

02:13

@Dhruv I did complex algebraic geometry (using differential geometry rather than algebra, mostly).

nickbros123

03:50

I heard from a senior that if you don't read calculus on manifold by Spivak after doing his calculus (1 variable) book then you'll forget the 1st book

Ted Shifrin

04:15

Total nonsense.

And there are far better books for the second.

nickbros123

05:58

I will trust you @TedShifrin. Is manifold calculus similar to multivariable calculus (sorry if I should ignorant), or is it a superset of it

Ted Shifrin

It is sophisticated multivariable calculus/linear algebra with more analysis and multilinear algebra thrown in.

insipidintegrator

07:05

Is (x,y) belongs to (RXZ)U(ZXR) a sufficient mathematical representation of the supposition that at least one of x, y is an integer?

leslie townes

sure, although many human beings would prefer the human language formulation. both are equally 'mathematical.'

nickbros123

08:04

@TedShifrin are you by chance the same person taught the math 3500 3510/10 series that's there on YouTube?

PM 2Ring

08:43

@nickbros123 The YouTube link to his lecture videos is in Ted's profile. math.stackexchange.com/users/71348/ted-shifrin

Peter

12:57

What is the smallest prime factor of this number ?

CroCo

13:11

@Yai0Phah after some reading, it seems to me the 1/2 comes from transforming a quadratic form into a symmetric matrix; hence, Q in this form must be symmetric, right? I come across different forms for the quadratic optimization problems. It is confusing.

Also, from linguistic perspective, I don't know what is the difference between canonical vs standard forms?

For a while I thought the two words are synonymous from mathematical point of view.

Peter

13:49

The above number has no prime factor upto $4\cdot 10^{10}$

Yai0Phah

14:18

@CroCo Conceptually, at your stage I guess, you should think of quadratic forms on a real vector space $V$ as a real valued function $q\colon V\to\mathbb R$ which is "quadratic" (i.e., under any choice of base, it can be written as a quadratic function on coordinates). If you have two Q's giving rise to the same function, then there is only one form.

nickbros123

15:03

When I expand ncr (the binomial coefficient) using the Pascal triangle formula, I'm able to do it indefinitely and the expansion looks like ( when I do it 4 times) (n-4)c(r-4)+4(n-4)c(r-3)+6(n-4)c(r-2)+4(n-4)c(r-1)+(n-4)cr. I understand this is just a consequence of the the Pascal triangle law, but is there another reason why I'm seeing the binomial coefficients again in the expansion of ncr, for a given number?

Ted Shifrin

16:37

@nickbros123 yes. How many times will you get $x^r$ when you expand $(1+ x)^n$?

nickbros123

@TedShifrin ncr times, i got the picture

My math teacher in highschool said, if you truly understand permutations, you will be able to understand why binomial coefficients are the way they are

That's the reason I haven't looked at the proof yet

amWhy

18:44

@TedShifrin Sorry, but Argentina beat France in the World Cup! I say this because your a self-admitted francophile? Or something like that, though I'm clueless as to whether you were following the World Cup, or if it mattered to you, or, if it did, who you were rooting for. :)

copper.hat

i was rooting for messi & mbappe

user4539917

19:00

France didn't even have one shot on net the whole first half?

👟⚽ 🥅 and then to come back from 2-0 to choke in the shoot out 🤯

🤔

copper.hat

19:20

yep, in a way it was a pity it was decided by penalties. howver, an awesome game.

and good solid play, no f*ckery

user4539917

🙏

That one ranks up there with the Giants ending the perfect season.

user4539917

20:24

> The 2026 men’s World Cup will be the first edition to feature 48 teams

16 more teams

Magnus Alexander

How many sylow 3-subgroups there are in $S_6 \times \mathbb{D}_{18}$? I've never dealt with products in those tasks, maybe someone can give me a hint

amWhy

21:04

Agreed, @copper.hat

Ted Shifrin

@amWhy I am indeed a Francophile, but not a soccer fanatic. Nevertheless, I did check the results while I was waiting for the Farmers Market to open at 10 AM. :)

amWhy

@TedShifrin Hah! Good enough for me!

@TedShifrin In my town, we don't have Farmers' Markets in December. :(

Ted Shifrin

Well, I drive about 50 miles round trip to go to this one (perhaps a bit excessive, but I have my favorites). Last week we had torrents of rain and wind, and I still went, but most of the vendors did not, so it was disappointing.

amWhy

But we will have a Winter Jamboree in January!!! :)

Goku

I'm actually pissed off France lost

At least I won my soccer game

Ted Shifrin

21:11

Much as I have my favorites in tennis (the only sport I follow with passion), I don't really get pissed off — even when Djokovic, whom I despise, wins.

amWhy

@Goku pobrecito!

@TedShifrin You show the spirit of sportsWo/Man/ship

Goku

@amWhy そうーナンーです

Darn I messed up the kanji

amWhy

@Goku I spoke in the language of Argentina. Your response needs to be in French.

Goku

@amWhy oh, I actually didn't know Argentina spoke Spanish since Google translate told me its a Spanish word

Ted Shifrin

21:37

Except for Portugal and Brazil, Spanish is a safe guess.

PM 2Ring

Some people in Argentina speak Welsh.

Y Wladfa

Y Wladfa (Welsh pronunciation: [ə ˈwladva], "The Colony"), also occasionally Y Wladychfa Gymreig (Welsh pronunciation: [ə wlaˈdəχva ɡəmˈreiɡ], "The Welsh Settlement"), refers to the establishment of settlements by Welsh immigrants in Patagonia, beginning in 1865, mainly along the coast of the lower Chubut Valley. In 1881, the area became part of the Chubut National Territory of Argentina which, in 1955, became Chubut Province.In the 19th and early 20th century the Argentine government encouraged emigration from Europe to populate Patagonia which, until the Conquest of the Desert began in the 1870s...

shintuku

a lot of diasporas in argentina

leslie townes

don't mention the war

shintuku

fun fact, before ww1 argentina's gdp per capita was higher than canada's

lots of european immigration from that era

lots of german speakers, lots of italian speakers

in fact in you tread around buenos aires you'll notice people speak with their hands a lot, like italians

also, the buenos aires argentine dialect has the same melodic rythm as italian

Thorgott

21:59

@MagnusAlexander a Sylow-group in a product is a product of Sylow-groups of the factors.

start by showing that the image of a $p$-Sylow under a surjective homomorphism is a $p$-Sylow and apply this to the projections onto the factors

Master

23:07

Say I have a probability density function $f$, and a point $\alpha$ whereby $f(\alpha) = 0$. Can I say anything probabilistically about $\alpha$

$f$ is not equal to 0 on an interval about $\alpha$, but solely at the point $\alpha$ itself

D Left Adjoint to U

@shintuku still down for studying after x-mas? I would pick either Steve Awodey's cat theory or JS Milne's AG.

shintuku

down for studying after the 21st

D Left Adjoint to U

K, what book?

We need a book mon. :D

Master

Can I say that my random variable $X$ with density $f$ will never take on a value $\alpha$

shintuku

nathan carther's visual group theory has an entire semester's lecture up on youtube with the course's reading schedule

Ted Shifrin

23:12

What’s your definition of a pdf, @Master?

D Left Adjoint to U

@shintuku do you have a link to the videos or book?

shintuku

math.clemson.edu/~macaule/classes/f22_math4120

D Left Adjoint to U

There's lots that come up on google, but not sure which one to click

shintuku

there's the class page

here's the youtube playlist

youtube.com/playlist?list=PLwV-9DG53NDxU337smpTwm6sef4x-SCLv

D Left Adjoint to U

@Shaun want to get down on this study group with us?

It's in your topic area: GT.

Master

23:14

@TedShifrin derivative of the cdf

shintuku

it's abstract algebra but half of it is group theory leading up to galois theory

D Left Adjoint to U

@shintuku That is great. Can never get enough group theory knowledge :)

Ted Shifrin

So this is a continuous RV?

Master

yes

Ted Shifrin

Then there’s never a positive probability of taking a particular value.

D Left Adjoint to U

23:17

@shintuku It's a study group for studying... groups. LOL :)

The recursion here is promising

Master

Yeah I know that, it's just can I say that the probability of taking values in an $\epsilon$ neighbourhood of this point is almost surely $0$

Ted Shifrin

Can you field a team, @DLeftAdjointtoU?

D Left Adjoint to U

not sure

what do you mean?

Ted Shifrin

I’m continuing the pun …

Master

What if my pdf was $f(x) = |x-\alpha|$ $x \in [\alpha -1, \alpha + 1]$

shintuku

23:18

heheheh

D Left Adjoint to U

oh, lol

Master

and 0 otherwise

shintuku

alternatively, there a full set of lectures on Artin's Algebra

on youtube

D Left Adjoint to U

Either one I'm happy with

shintuku

with course page here: web.archive.org/web/20091114060618/http://…

D Left Adjoint to U

23:19

it's up to you guys

Master

Can I say anything special about the values close to $\alpha$

Ted Shifrin

So the integral is positive over any tiny interval.

Master

Yeah

The integral represents the probability of being in that interval

shintuku

Ted, ever heard anything about some abstract algebra lectures on youtube?

D Left Adjoint to U

@shintuku I don't like how that page is archived though. If you could find one that's not archived, that would be greater

shintuku

23:22

yeah the lectures are from 2003

Ted Shifrin

The cdf has an inflection point at $\alpha$, nothing more, right?

D Left Adjoint to U

I started with Michael Artin's algebra a long time ago, it was a decent book

Master

I guess the best I can say is the probability of achieving values close to $\alpha$ is smaller than any other $\epsilon$ neighbourhood of another point.

shintuku

@DLeftAdjointtoU the corresponding set of lectures is here youtube.com/…

Ted Shifrin

@shintuku You talking about Ben Gross lectures at Harvard?

Artin is a great book.

Master

23:23

I am referencing this paper: https://www.rand.org/content/dam/rand/pubs/research_memoranda/2006/RM408.pdf
page 8 and 9.

shintuku

at Ted: yeah I just found the Ben Gross lectures on youtube

we're looking for material to start an abstract algebra study group

Master

The centre point of the hexagon is assigned density $0$, I know it's a continuous variable but I want to say something mathematically correct along the lines that it is impossible that we play the centre strategy

D Left Adjoint to U

@shintuku Either of those courses is fine. Ben Gross's seems more advanced

We'll start a chat room I guess, when the time comes

shintuku

we've got Ted's recommendation on Artin, so i'd say we pick that class

Master

In this thread: stackoverflow.com/questions/5148744/… the person answering says there is $0$ probability of obtaining the centre point. Is this correct?

D Left Adjoint to U

23:28

@shintuku know of a link to the book?

Z-library died because of obvious legal reasons

I bought that book a long time ago, don't know what happened to it. Should I have to buy it again?

Ted Shifrin

Artin is the most mathematically interesting undergrad algebra course I know. Excellent problems, too. In particular, he integrates linear algebra material throughout. Totally unique in that respect.

3

Back to the kitchen.

shintuku

thanks for the comment Ted

Transcript for

$Mathematics$ Mathematics